@Joe, I don't have enough info to compose a full answer but I'd like to bring up the case of Pytheas who is unfortunately known to us only by second hand relations (as is Himilco). One of the most interesting references on Pytheas is Barry Cunliffe's book (0140297847) and there is a fairly detailed discussion of latitude determination - and some evaluation of its precision - in wikipedia's article here‌​.
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Alain PannetierJul 11 '12 at 8:52

Some information about ancient latitude/longitude navigation system is in the 1421 book by Gavin Menzies. However, information in this book is certainly to be taken with caution, and I do not have my copy offhand.
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Jean-Christophe DubacqJul 13 '12 at 19:17

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Ben Crowell says 15 minutes of latitude in the Almagest of Ptolemy according to Goldstein. Along these lines I examined some ancient sources and have the following findings:

In the Geography of Ptolemy, it reads "The fourth parallel is distant one hour [from the equator] and is 16°25'. This is parallel [latitude] through Meroe. Actually Meroe lies between 16°53' and 17°00', so Ptolemy is only accurate to the nearest degree on this measurement. However, since this part of the Geography is for making globes, greater than one degree accuracy is not necessary.

In his measurement of the island of Capri, Ptolemy gives 40°10' latitude and 39°20' longitude. In our measurement, the island is at 40°32' latitude, and 14°11-16' longitude. So, he is too small in latitude by 22 minutes. Note that Ptolemy gives measurements in increments of 5 minutes (claimed accuracy).

Now, let us consider Book IV of the Geography, Chapter V, Egypt. Ptolemy gives the coordinates of Heliopolis as 29°50' latitude and 62°30' longitude, whereas our modern measurement is 30°07' latitude and 31°18' longitude. So the latitude measurement is too small by 17 minutes.

Therefore, we see in one measurement he is -22 minutes and the other -17 minutes, which are 5 minutes apart. Hence, Ptolemy's claim of 5 minute precision seems to be roughly correct. He has systematic error of all his latitude measurements being about 20 minutes too low, and a measurement error of about 5 minutes, which is his claimed precision. More study would be required to determine whether his -20 minute error was universal, or whether other astronomers made the same error, based on an inaccurate measurement of the earth, perhaps.

Now, let us consider longitude. Between Capri and Heliopolis, Ptolemy measures 23°10' by subtraction from the values given above. Our modern difference in longitude is 17°2-7'. There is a difference of 6 degrees.

To see if there is a systematic error, let us look at another measurement, that of Caesarea Strotonis, the Roman capital in Palestine, from where Pontius Pilate held forth. Today we measure its longitude as 34°53' being 20°40' distant from Capri. Ptolemy gives the longitude as 60°15' which is 20°55' from his measurement of Capri, only a 15 minute difference from the modern measurement. So, it is interesting that in some longitude measurements he is accurate to within minutes, but in others off by over 5 degrees.

From this we can see that, at least based on these examples there is no systematic bias in his longitude measurements, but an accuracy difference of up to 6 degrees longitude. There does appear to be a bias in his latitude measurement of 20 minutes, and if we adjust for this bias, then he comes within 5 minutes of latitude consistently.

There is a book, History and Practice of Ancient Astronomy by James Evans, which goes into great detail on the methods of the ancients. Unfortunately, he does not specifically try to characterize the accuracy of the measurements, but after reading through this book, the impression I get is that the ancient Greeks after Hipparchus were capable of measuring to the nearest 1 minute of latitude and 2 degrees of longitude.

Latitude can be calculated from observations of stellar objects (typically using something like an astrolabe) and a bit of math. The Greeks could do this as early as 150BC, but only on dry land. The Mariner's Astrolabe wasn't invented until around 1300 CE.

Nobody had a good way to determinte longitude in realtime aboard a ship before the invention of the marine chronometer in the early 1700's. The closest anyone came was the Chinese, who managed to work out the longitude of various places on the Indian trade routes in 1421 by placing observers on said places to observe various lunar and stellar positions simultaniously. This information may have made their maps better, but wasn't particularly useful to a navigator out of sight of land.

Before that, the typical technique used was dead reckoning, which was incredibly inaccurate. Basically, the navigator would chuck a hunk of wood out the back of the ship, try to estimate their speed based on their relative speed to the jetsam, and try to calculate their distance from the last time they did that based on that speed. Obviously this doesn't take currents into account at all, and any errors are likely to accumulate every time you do it.

What was typically done in the Medeteranian in ancient times was that navigators just kept themselves in sight of land. Even then, bad things could happen. For instance, the Odyssey is essentially a story of an ancient Greek who got blown off course sailing home from nearby Anatolia, and spent 10 years trying to find his way home.

I'm aware of how you calculate latitude, but what I'm looking for is the accuracy. How accurately could the Greeks determine their latitude?
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JoeJul 9 '12 at 22:19

I don't think it was possible to get accurate readings if they believed the Earth flat. Also, did they have the concept of this system, anyway?
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slyblotyJul 10 '12 at 13:22

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@slybloty - That's kind of a myth. The ancient Greeks realized the earth was spherical (and came up with a pretty good estimate of its size), and any mariner could look at the curved horizon and see it with their own eyes.
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T.E.D.♦Jul 10 '12 at 13:27

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@T.E.D. You might know that the earliest known Greek traveler to the English Isles is supposed to be Pytheas (it is as you may know the etymology of Britain, which suggests that Celts were probably as tattooed as the Picts themselves). Pytheas seems to have gone much further Northwards than Britain actually, was apparently enrolled for his mathematics skills and, more to the point - that was around 330/300 BC - had developed several ways to calculate latitude. Regarding the hunk of wood: that's the etymology of our...
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Alain PannetierJul 10 '12 at 21:30

To find the latitude of a point on the land, one would simply have to measure the elevation of Polaris above the horizon. Therefore the question of the precision of (land-based) latitude determinations in this period reduces to the question of how accurately people could measure angles in the sky. The Almagest was the state of the art during this time, and its angular measurements seem to have been precise to about 15-30 minutes of arc.[Goldstein 1976]

Longitude

Since there were no accurate clocks until Galileo, and no sea-transportable accurate clocks until much later, the determination of a longitude in the ancient world would have been equivalent to estimating an east-west distance (using surveying chains, estimates of sailing speeds, ...) and dividing by the size of the earth. Columbus appears to have underestimated the size of the earth by about a factor of 2, leading to his belief that he could reach China and Japan by sailing across the Atlantic. So even as late as the Italian renaissance, the conversion factor between longitude and distance appears to have been uncertain by about a factor of 2. This is very roughly comparable to the precision of the ancient estimate of the radius of the earth by Eratosthenes, i.e., there was not much improvement over 1700 years.

First, a spot of background science. The Longitude Problem is exactly identical to the problem of establishing simultaneity on widely separated locations on the Earth's surface, and both prerequisite the existence of a reliable estimate of the Earth's diameter. Certainly Eratosthenes calculated the Earth's diameter in the 3rd Century BC, and other civilizations may have done so by roughly the same time period. However the problem of establishing simultaneity is more difficult, and comes in two flavours.

Longitude is calculated by comparing the elevation of an astronomical object to the pre-calculated (or observed) elevation of the same object at a reference location at the precisely simultaneous moment in time. Everything in the sky rotates once around that vast celestial sphere every 24 hours, so the more precisely one can establish simultaneity the more precise one's measurement of longitude will be.

The problem is simpler when the goal is cartography - calculating the longitude, and thus precise location, of a given spot on the globe exactly once. In this case one can use the occurrence of a predicted astronomical event as the definition of simultaneity. Surveying teams are organized to travel to the specified locales well in advance of the event, and providing the skies are clear on the given day the necessary observations are made. Once the surveying teams return the results are tabulated and the maps drawn.

The more difficult problem, and the one that confounded the British Admiralty into establishing the Longitude Prize, is of establishing the location of a moving vessel out of sight of land at whatever time the skies happened to be clear, wherever and whenever that was. One could not halt a sailing vessel in the middle of the ocean and wait for a pre-calculated event that occurred once or twice a month at best. It was necessary to resort to Dead Reckoning, a well-established and remarkably accurate science by the 17th and 18th centuries, which provided locations within one or two dozen miles on voyages thousands of miles long. When the goal was simply to voyage out and return home, this was more than adequate. However when the need is to avoid reefs of only a few hundred yards extent, being off by a few miles all too often results in foundering instead of sailing safely by.

The accuracy of dead reckoning can be judged by the quality of 16th and 17th century maps, reproductions of which are readily available all over the web. Don't be misled by the contours of western North America - those are due to the wanderings of the North Geomagnetic Pole.

To measure latitude, you must measure the elevation of some celestial body. Basically, you will use the the Sun, or stars (trajectory of planets and the Moon is too complex to serve much here).

If you use the Sun, then you are using a projected shadow (you don't look at the Sun directly). You have some big pole, that you try to erect as vertical as possible; and you measure the length of the shadow at noon on the equinox. You will need to do some year-long measures to work out when the equinox actually is. The crucial point is that the Sun is not a point in the sky; it has an apparent diameter of about 30' (half a degree). This is the reason why, when you look at the shadow of some building, projected on the floor, the shadow edge is blurry: this transition zone between shadow and non-shadow corresponds to the floor spots from which the Sun is partially visible, and partially hidden by the building. The bottom-line is that a Sun-based measure of latitude tends to be imprecise: accuracy is within half a degree, but no better. (With a sextant you can have a much better accuracy, but that's because that apparatus includes filters allowing the operator to actually look at the Sun, and target the disc edge, instead of "the Sun in general" as in a shadow-based measure.)

With stars you can do potentially better, because they are points (at least to the naked eye) and you can stare at them directly without going blind. If you use stars, then you must follow several for a night, noting their azimuth and elevation throughout the night: this is sufficient to recompute their apparent trajectory, and then work out the latitude. The accuracy of the human eye is, at best, 1' (1/60th of a degree). However, it is hard to achieve in practice.

Notably, even if you can see a 1' angle deviation, the measure will depend on the accuracy with which you know the geometric characteristics of the device you are using (including the measure of "vertical" and "horizontal"). Also, prior to Gauss and Legendre in the early 19th century, astronomers had no systematic method to deal with measure errors and smooth them out with averages and statistics.

As a data point, Tycho Brahe, in the late 16th century, achieved measures with about 2' accuracy on average. These measures would translate, indeed, into a computation of latitude with the same accuracy. It must be noted that Brahe had very good eyesight, was exceptionally stubborn, and benefited from the precision offered by late Renaissance instruments when it came to measuring the length of, say, a ruler (according to David S. Landes, we have to thank clockwork technology for the availability of such tools in the Renaissance).

As another data point, the Great Pyramid of Giza (built circa 2560 BC) is aligned on the cardinal point within 4'.

From all these informations, we may conclude that astronomers from around 1 AD could achieve a measure of latitude with an accuracy of about 4' or so, but at considerable expense. Hipparchus has apparently done so in some occasions, but he had dedicated his life to such matters.

Longitude

Longitude is much harder: it can be measured by the difference between the local time and a reference time. If the Sun appears to reach noon while your watch says 2 o'clock (while it matched the Sun in your home town), then you know that you went 30 degrees to the West. This is about the only direct way to measure longitude: you need to bring a clock with you, and you will get as much accuracy as your clock provides, with 1 degree of longitude for every 4 minutes of time. Since clocks in antiquity were awfully inaccurate, this was not workable at that time. Indeed, longitude is measured by the difference in time between a sundial (which measures local time) and a clock (set to the reference time). When the clock is less accurate than a sundial, it is quite hard to reach any conclusion at all.

Some indirect measures can be done in rare occasions but need astronomical devices that were not available at that time (e.g. telescopes to observe the transit of Venus when it passes between the Sun and the Earth).

All longitude estimates in antiquity use the indirect method by which the longitude is inferred from the actual land distance, obtained through some other mean (mostly triangulation with notable geographical features such as hills and buildings). This works well with short distances (e.g. between Athens and Corinth), much less for long distances, and very badly when a sea is involved. Contrary to latitude, astronomers in antiquity could not get a notion of absolute longitude, only relative for locations which are close enough to each other.

Another lesson on methodology. The OP already knows how to calculate celestial coordinates. They asking about the accuracy of the measurement in ancient times, not a tutorial on how to do it.
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Tyler DurdenSep 13 '14 at 15:31

The Greek astronomers (e.g. Ptolemy) could calculate longitude and latitude using spherical trigonometry. Their calculations are accurate on the assumption that the Earth is a perfect sphere. Our astronomers today believe that the Earth is slightly pear-shaped and consequently arrive at a slightly different calculation of longitude and latitude.

The pear-shape of the earth isn't very relevant to ocean navigation, as it is in feet relative to thousands of miles. Latitude is easy, longitude, not so much.
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OldcatMar 29 '14 at 0:41

@fdb At the risk of sound pedantic, whether or not the average Greek knew the Earth was not flat depends on the time period in question. Virtually all of the pre-Socratics believed in a flat-Earth cosmology.
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David HMar 29 '14 at 4:08